UNIT2: LOGARITHMIC AND EXPONENTIAL FUNCTIONS
UNIT2: LOGARITHMIC ANDEXPONENTIAL FUNCTIONS
UNIT2: LOGARITHMIC AND EXPONENTIAL FUNCTIONS
Key unit competence
Extend the concepts of functions to investigate fully logarithmic and exponential
functions and use them to model and solve problems about interest rates, populationgrowth or decay, magnitude of earthquake, etc.
Introductory activity
The Accountant for a Health Center receives money from patients in an interesting
way so that the money he/she earns each day doubles what he/she earned the
previous day. If he/she had 200USD on the first day and by taking t as the number
of days, discuss the money he/she can have at the tth day through answering the
following questions:
a) Draw the table showing the money this Health Center Owner will have on
each day starting from the first to the 10th day.
b) Plot these data in rectangular coordinates
c) Based on the results in a), establish the formula for the Health Center
Owner to find out the money he/she can earn on the nth day. Therefore, if t
is the time in days, express the money F (t ) for the economist.
d) Now the Health Center Owner wants to possess the money F under the
same conditions, discuss how he/she can know the number of daysnecessary to get such money from the beginning of the business.
From the discussion, the function F (t ) found in c) and the function Y (F ) found in
d) are respectively exponential function and logarithmic functions that are needed
to be developed to be used without problems. In this unit, we are going to study the
behaviour and properties of such essential functions and their application in real lifesituation.
2. 1 Logarithmic functions2.1.1 Domain of definition for logarithmic function
If the base is 10, it is not necessary to write the base, and we say decimal logarithm
or common logarithm or Brigg’s logarithm. So, the notation will become y = log x . If
the base is e (where e =2.718281828…), we have Neperean logarithm or naturallogarithm denoted by y = ln x instead of loge y = x as we might expect.
2.1.2 Limits and asymptotes of logarithmic functions
Activity 2.2
Consider the form of this graph then by using calculator, complete the table below to answer the
questions that follow.
2.1.3 Continuity and asymptote of logarithmic functions
• The logarithmic function is increasing and takes its values (range) from
negative infinity to positive infinity.
Example 2.3
Let us consider the logarithmic function y = log2 (x − 3)
a) What is the equation of the asymptote line?
b) Determine the domain and range
c) Find the x − intercept.
d) Determine other points through which the graph passes
e) Sketch the graph
Solution
a) The basic graph of 2 y = log x has been translated 3 units to the right, so the line L ≡ x = 3
is the vertical asymptote.
b) The function y = log2 (x − 3) is defined for x − 3 > 0
So, the domain is ]3,+∞[ .The range is
c) The intercept is (4, 0) since log2 (x − 3) = 0⇔ x = 4
d) Another point through which the graph passes can be found by allocating an arbitrary value
to x in the domain then compute y.
For example, when x = 5, y = log2 (5 − 3) = log2 2 =1 which gives the point (5,1) .
Note that the graph does not intercept y-axis because the value 0 for x does not belong tothe domain of the function.
2.1.4. Differentiation of logarithmic functions
2. 3 Applications of logarithmic and exponential functions
Logarithmic and exponential functions are very essential in pure sciences, social
sciences and real life situations. They are used by bank officers to deal with interests
on loans they provide to clients. Economists and demographists use such functions
to estimate the number of population after a certain period and many researchers
use them to model certain natural phenomena. We are going to develop some ofthese applications.
When a person gets a loan (mortgage) from the bank, the mortgage amount M, the
number of payments or the number t of years to cover the mortgage, the amount
of the payment P, how often the payment is made or the number n of payments peryear, and the interest rate r, it is proved that all the 5 components are related by
Example 2.12
A business woman wants to apply for a mortgage of 75,000 US dollars with an
interest of 8% per month that runs for 20 years. How much interest will she pay overthe 20 years?
b) At the beginning, (t = 0), the number of fish is 500. After 4 months (t = 4),
the number of fish is 4,1048.
c) As the time x increases, the number of fish will be 10,000.
d) The population is increasing most rapidly after 4 months. This is becausethe increment of fish after 1 month is greater.
b) After 5 days, the calculator and the graph show that 54 students will be
infected.
c) According to this model, when the time increases without bound, the graph
shows that all students can be infected. However, in real life, the infinite
time is not possible. Therefore, all students cannot be infected.
2.3.5 Earthquake problems
Activity 2.15
Do the research in the library or explore internet to find out how Charles Richter
tried to compare the magnitude of two earthquakes by the use of logarithmicfunction.
Seismographic readings are made at a distance of 100 kilometers from the epicenter
of an earthquake. If there is no earthquake, the seismographic reading is x0 = 0.001millimeter.
Example 2.17
A scientist determines that a sample of petrified wood has a carbon-14 decay rate of
8.00 counts per minute per gram. What is the age of the piece of wood in years? The
decay rate of carbon-14 in fresh wood today is 13.6 counts per minute per gram, andthe half- life of carbon-14 is 5730 years.
END UNIT ASSESSMENT
